Strange drawing of selections in edit mode
When a selection looks like that it usually means that there is a double vertex there. Two vertices in the same spot, and you only selected one of them.  

To select both vertices, change the Viewport Shading mode to Wireframe Z, then with either circle select C, border select B, or just right clicking twice, select both vertices. If you want only one vertex there, with both (or more) selected press Alt M > At Center.